Warm upWarm up1. Find the

solutions.

2. Find the interval of decrease.

(-1, 0) (5, 0)

x > 2

Warm-upFind the slope

y

xRun=6-2=4

Rise=3-1=2

= Rise

Run

2 1

4 2

(2,1)

(6,3)

Constant Rate of ChangeConstant Rate of Change

The The slopeslope of a nonvertical line is the of a nonvertical line is the ratio of the vertical change (rise) to ratio of the vertical change (rise) to the horizontal change (run) between the horizontal change (run) between any two points on the line.any two points on the line.

Find the slope between (2, 4) and (4, Find the slope between (2, 4) and (4, 8).8).

2 1

2 1

y ychange in ym

changein x x x

Rate of ChangeRate of Change

The The rate of changerate of change is the ratio of the is the ratio of the change of one quantity to a change in change of one quantity to a change in another quantity.another quantity.

Example:Example:

- The table shows the amount of water - The table shows the amount of water evaporating from a swimming pool on a evaporating from a swimming pool on a hot day. Find the rate of change in hot day. Find the rate of change in gallons with respect to time.gallons with respect to time.Time (hours) 2 6 12

Gallons evaporated 4.5 13.5 27

Rate of ChangeRate of Change

Where is the greatest rate of Where is the greatest rate of change on the graph? What is the change on the graph? What is the value?value?

Partner PracticePartner Practice

Calculating rate of change from a Calculating rate of change from a table.table.

Example 3Example 3

Example 4Example 4

2.09 1.990.10

2 1dollars per day

43 382.5 deg

2 0rees per hour

ExtensionExtension

Average Rate of ChangeAverage Rate of Change

The The average range of changeaverage range of change between any two points (xbetween any two points (x11,f(x,f(x11)) and )) and (x(x22,f(x,f(x22)) is the slope of the line )) is the slope of the line through the 2 points.through the 2 points.

2 1

2

2

1

1

2 1

( ) ( )y ym becomes

x x

f x f x

x x

Example 1Example 1

Find the average rate of change of Find the average rate of change of f(x) = 2xf(x) = 2x22 – 3 when x – 3 when x11= 2 and x= 2 and x22 = 4. = 4.

Example 2Example 2

Find the average rate of change of Find the average rate of change of f(x) = xf(x) = x33 – 3x when x – 3x when x11= -2 and x= -2 and x22 = 0. = 0.

Walking Task• To begin a class discussion of speed, Dwain and

Beth want to stage a walking race down the school hallway. After some experimentation with a stop watch, and using the fact that the flooring tiles measure 1 foot by 1 foot, they decide that the distance of the race should be 40 feet and that they will need about 10 seconds to go 40 feet at a walking pace.

• They decide that the race should end in a tie, so that it will be exciting to watch, and finally they make a table showing how their positions will vary over time. Your job is to help Dwain and Beth make sure that they know how they should walk in order to match their plans as closely as possible.

The Data

Time (seconds) 0 1 2 3 4 5 6 7 8 9 10

Dwain’s position (feet) 0 4 8 12 16 2

0 24 28 32 36 40

Beth’s position (feet) 0 1 3 6 10 15 20 25 30 3

5 40

Draw a graph. Should you connect the

dots?

Beth (blue)Dwain (Purple)

Describe how Dwain and Beth should walk in order to match their data.

Beth (blue)Dwain (Purple)

What is Beth and Dwain’s average speed from 0 to 3 seconds?

Beth (blue)Dwain (Purple)

What is Beth and Dwain’s average speed from 6 to 8 seconds?

Beth (blue)Dwain (Purple)

What is Beth and Dwain’s average speed from 0 to 10 seconds?

HomeworkHomework

Practice Worksheet and Coach Page Practice Worksheet and Coach Page 9494